Question

How do you evaluate ${\sec }^{-1}2?$

Answer 1

${\sec }^{-1}2=\frac{\pi }{3}$

Explanation:

Let the answer be x.

Since $se{c}^{-1}\left(2\right)=x,\sec \left(x\right)=2$.

As secant is the reciprocal of the cosine function, i.e. $sec\left(x\right)=\frac{1}{cos\left(x\right)},$ the following must be true as well.

$\frac{1}{cos\left(x\right)}=2$

Or $cos\left(x\right)=\frac{1}{2}.$

The range of $se{c}^{-1}\left(x\right)$ is $[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$.

The range of $co{s}^{-1}\left(x\right)$ is $[0,\pi ]$.

Since the ranges are similar, we can find x using

$x=co{s}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$